A test plane flies in a straight line with positive velocity $v(t)$, in miles per minute at time $t$ minutes, where $v$ is a differentiable function of $t$. Selected values of $v(t)$ for $0 \leq t \leq 40$ are shown in the table below.

\begin{tabular}{ c } $t$

(minutes)

& 0 & 5 & 10 & 15 & 20 & 25 & 30 & 35 & 40 \hline

$v(t)$

(miles per minute)

& 7.0 & 9.2 & 9.5 & 7.0 & 4.5 & 2.4 & 2.4 & 4.3 & 7.3 \hline \end{tabular}
(a) Use a midpoint Riemann sum with four subintervals of equal length and values from the table to approximate $\int_{0}^{40} v(t)\,dt$. Show the computations that lead to your answer. Using correct units, explain the meaning of $\int_{0}^{40} v(t)\,dt$ in terms of the plane's flight.
(b) Based on the values in the table, what is the smallest number of instances at which the acceleration of the plane could equal zero on the open interval $0 < t < 40$? Justify your answer.
(c) The function $f$, defined by $f(t) = 6 + \cos\left(\frac{t}{10}\right) + 3\sin\left(\frac{7t}{40}\right)$, is used to model the velocity of the plane, in miles per minute, for $0 \leq t \leq 40$. According to this model, what is the acceleration of the plane at $t = 23$? Indicate units of measure.
(d) According to the model $f$, given in part (c), what is the average velocity of the plane, in miles per minute, over the time interval $0 \leq t \leq 40$?

4. A squirrel starts at building $A$ at time $t = 0$ and travels along a straight, horizontal wire connected to building $B$. For $0 \leq t \leq 18$, the squirrel's velocity is modeled by the piecewise-linear function defined by the graph above.
(a) At what times in the interval $0 < t < 18$, if any, does the squirrel change direction? Give a reason for your answer.
(b) At what time in the interval $0 \leq t \leq 18$ is the squirrel farthest from building $A$ ? How far from building $A$ is the squirrel at that time?
(c) Find the total distance the squirrel travels during the time interval $0 \leq t \leq 18$.
(d) Write expressions for the squirrel's acceleration $a ( t )$, velocity $v ( t )$, and distance $x ( t )$ from building $A$ that are valid for the time interval $7 < t < 10$.

WRITE ALL WORK IN THE EXAM BOOKLET.

1. Consider the differential equation $\frac { d y } { d x } = \frac { x + 1 } { y }$.
   (a) On the axes provided, sketch a slope field for the given differential equation at the twelve points indicated, and for $- 1 < x < 1$, sketch the solution curve that passes through the point $( 0 , - 1 )$. (Note: Use the axes provided in the exam booklet.) [Figure]
   (b) While the slope field in part (a) is drawn at only twelve points, it is defined at every point in the $x y$-plane for which $y \neq 0$. Describe all points in the $x y$-plane, $y \neq 0$, for which $\frac { d y } { d x } = - 1$.
   (c) Find the particular solution $y = f ( x )$ to the given differential equation with the initial condition $f ( 0 ) = - 2$.
2. Two particles move along the $x$-axis. For $0 \leq t \leq 6$, the position of particle $P$ at time $t$ is given by $p ( t ) = 2 \cos \left( \frac { \pi } { 4 } t \right)$, while the position of particle $R$ at time $t$ is given by $r ( t ) = t ^ { 3 } - 6 t ^ { 2 } + 9 t + 3$.
   (a) For $0 \leq t \leq 6$, find all times $t$ during which particle $R$ is moving to the right.
   (b) For $0 \leq t \leq 6$, find all times $t$ during which the two particles travel in opposite directions.
   (c) Find the acceleration of particle $P$ at time $t = 3$. Is particle $P$ speeding up, slowing down, or doing neither at time $t = 3$ ? Explain your reasoning.
   (d) Write, but do not evaluate, an expression for the average distance between the two particles on the interval $1 \leq t \leq 3$.

WRITE ALL WORK IN THE EXAM BOOKLET. END OF EXAM

© 2010 The College Board. Visit the College Board on the Web: \href{http://www.collegeboard.com}{www.collegeboard.com}.

The graph below shows the distance traveled by a cyclist as a function of time.
Based on the graph, what is the average speed of the cyclist, in km/h?
(A) 10
(B) 15
(C) 20
(D) 25
(E) 30

181. The position–time graph of a moving object is shown below as a parabola. In the time interval 0 to $8\,\text{s}$, what are the magnitude of average velocity and average speed in SI?
[Figure: A parabolic position-time graph with $x(\text{m})$ on the vertical axis and $t(\text{s})$ on the horizontal axis. The curve reaches a maximum of 24 m, passes through 16 m, and returns to 0 at $t = 8\,\text{s}$.]

1. [(1)] 1 and zero
2. [(2)] 2 and zero
3. [(3)] 1 and 1
4. [(4)] 2 and 2

157. The velocity–time graph of a particle moving in a straight line is shown in the figure below. The distance traveled by this particle in the time interval $0\,\text{s}$ to $20\,\text{s}$ is how many meters?
\begin{minipage}{0.45\textwidth} [Figure: V(m/s) vs t(s) graph. The graph starts at $V=0$ at $t=0$, decreases to $V=-8$ at some time, then increases linearly to $V=22$ at $t=15\,\text{s}$, then decreases back to $V=0$ at $t=20\,\text{s}$.] \end{minipage} \begin{minipage}{0.45\textwidth}

• [(1)] $160$
• [(2)] $175$
• [(3)] $180$
• [(4)] $192$

\end{minipage}

48. Two moving objects start from the origin at the same time, and their velocity–time graphs are as shown in the figure. At the moment when the two objects move in opposite directions, how does the distance between them change?
\begin{minipage}{0.4\textwidth} [Figure: velocity-time graph showing two lines A and B; line A starts from $-6$ on the V-axis and increases; line B starts from $-4$ on the V-axis and increases with a different slope; time axis shows values 2 and 8 in seconds.] \end{minipage} \begin{minipage}{0.55\textwidth}

1. [(1)] Increases by 48 meters.
2. [(2)] Decreases by 48 meters.
3. [(3)] Increases by 64 meters.
4. [(4)] Decreases by 64 meters.

\end{minipage}

35. The phase space diagram for a ball thrown vertically up from ground is
(A) [Figure]
(B) [Figure]
(C) [Figure]
(D) [Figure]

ANSWER: D

1. The phase space diagram for simple harmonic motion is a circle centered at the origin. In the figure, the two circles represent the same oscillator but for different initial conditions, and $\mathrm { E } _ { 1 }$ and $\mathrm { E } _ { 2 }$ are the total mechanical energies respectively. Then
   (A) $\quad E _ { 1 } = \sqrt { 2 } E _ { 2 }$
   (B) $E _ { 1 } = 2 E _ { 2 }$
   (C) $E _ { 1 } = 4 E _ { 2 }$
   (D) $E _ { 1 } = 16 E _ { 2 }$ [Figure]

ANSWER: C

1. Consider the spring-mass system, with the mass submerged in water, as shown in the figure. The phase space diagram for one cycle of this system is
   (A) [Figure]
   (B) [Figure]
   (C) [Figure]
   (D) [Figure]

ANSWER: B

Paragraph for Question Nos. 38 and 39

A dense collection of equal number of electrons and positive ions is called neutral plasma. Certain solids containing fixed positive ions surrounded by free electrons can be treated as neutral plasma. Let ' $N$ ' be the number density of free electrons, each of mass ' $m$ '. When the electrons are subjected to an electric field, they are displaced relatively away from the heavy positive ions. If the electric field becomes zero, the electrons begin to oscillate about the positive ions with a natural angular frequency ' $\omega _ { \mathrm { p } }$ ', which is called the plasma frequency. To sustain the oscillations, a time varying electric field needs to be applied that has an angular frequency $\omega$, where a part of the energy is absorbed and a part of it is reflected. As $\omega$ approaches $\omega _ { \mathrm { p } }$, all the free electrons are set to resonance together and all the energy is reflected. This is the explanation of high reflectivity of metals.

A person climbs up a stalled escalator in 60 s. If standing on the same but escalator running with constant velocity he takes 40 s. How much time is taken by the person to walk up the moving escalator?
(1) 37 s
(2) 27 s
(3) 24 s
(4) 45 s

The velocity $(v)$ and time $(t)$ graph of a body in a straight line motion is shown in the figure. The point $S$ is at 4.333 seconds. The total distance covered by the body in 6 s is:
(1) $\frac{37}{3}$ m
(2) 12 m
(3) 11 m
(4) $\frac{49}{4}$ m

The speed verses time graph for a particle is shown in the figure. The distance travelled (in $m$) by the particle during the time interval $\mathrm { t } = 0$ to $\mathrm { t } = 5$ s will be $\_\_\_\_$

From the $v - t$ graph shown, the ratio of distance to displacement in 25 s of motion is: [Figure]
(1) 1
(2) $\frac{1}{2}$
(3) $\frac{5}{3}$
(4) $\frac{3}{5}$

Q2. A particle moving in a straight line covers half the distance with speed $6 \mathrm {~m} / \mathrm { s }$. The other half is covered in two equal time intervals with speeds $9 \mathrm {~m} / \mathrm { s }$ and $15 \mathrm {~m} / \mathrm { s }$ respectively. The average speed of the particle during the motion is :
(1) $10 \mathrm {~m} / \mathrm { s }$
(2) $8 \mathrm {~m} / \mathrm { s }$
(3) $9.2 \mathrm {~m} / \mathrm { s }$
(4) $8.8 \mathrm {~m} / \mathrm { s }$

Two light spots move on a straight track of length 120 cm. When they reach an endpoint, they reverse direction and continue moving. Initially, the two spots are at the two ends of the track moving toward each other. Light spot A and light spot B move at speeds of 5 cm/s and 10 cm/s respectively. Select the correct options.
(1) The position of the first meeting of the two light spots is 40 cm away from one of the endpoints
(2) The position of light spot A exhibits periodic behavior with a period of 24 seconds
(3) When light spot A returns to its starting point, light spot B is also at its starting point
(4) The second meeting of the two light spots occurs at one of the endpoints
(5) There are 3 different meeting positions for the two light spots on the track

In the graph below showing Fatih's time to reach work according to his departure time from home on a certain day, the graph representations between 07.00-08.00 and 08.00-09.00 are linear.
Fatih, who left home at some time between 08.00 and 09.00, would have arrived at work at the same time if he had left home exactly one hour earlier.
Accordingly, at what time did Fatih arrive at work?
A) 09.12 B) 09.15 C) 09.18 D) 09.21 E) 09.24